The varieties of characters of representations of finitely generated groups into SL(2,C) have interesting relations to
knot theory, spectral geometry and geometric quantization and their study goes back to the work of Poincaré on the
monodromy of linear second order differential equations. We will survey some of the main recent developments in this
theory, and show how the theory of conjugation invariants of n-tuples of 2 by 2 matrices can be used to clarify and
improve them.